# Eikonal Blog

## 2010.01.26

### A Theorem of Ellis and Pinsky

Filed under: mathematics — Tags: — sandokan65 @ 13:12

Theorem (Ellis & Pinsky): Let $\latex A$, $B$ be real symmetric $n\times n$ matrices, and assume that $B$ is negative semi-definite. Then as $\epsilon \rightarrow +0$:

• $e^{t(A+i\frac1{\epsilon}B)} e^{-i\frac{t}{\epsilon}B} = e^{t A_0} + {\cal O}(\epsilon)$,

where $A_0 :\equiv \sum_{\lambda\in\sigma(B)} P_\lambda A P_\lambda$, here $P_\lambda$ being the orthogonal projector on the eigenspace of $B$ associated with the eigenvalue $\lambda$.

Source: T1270 = S.L.Campbell “On the limit of a product of matrix exponentials”; Linear and Multilinear Algebra; 1978, Vol 6, p.p. 55-59.